This issue of "Note di Matematica" is dedicated to the memory of Bruno Moscatelli, Editor in Chief of "Note di Matematica", until his death in 2008. Several authors and friends contributed with papers about various subjects of Mathematical Analysis.

It is shown that the generator of every exponentially equicontinuous, uniformly continuous $C_0$--semigroup of operators in the class of quojection Fréchet spaces $X$ (which includes properly all countable products of Banach spaces) is necessarily everywhere defined and continuous. If, in addition, $X$ is a Grothendieck space with the Dunford--Pettis property, then uniform continuity can be relaxed to strong continuity. Two results, one of M. Lin and one of H.P. Lotz, both concerned with uniformly mean ergodic operators in Banach spaces, are also extended to the class of Fréchet spaces mentioned above. They fail to hold for arbitrary Fréchet spaces.

In [12] Piszczek proved that "tameness always implies quasinormability" in the setting of Fréchet spaces. In this paper we present an alternate proof of this interesting result.

We study the analyticity of the semigroups generated by some classes of degenerate second order differential operators in the space of continuous function on a domain with corners. These semigroups arise from the theory of dynamics of populations.

We solve a long standing open problem, by proving the analyticity of the semigroups generated by a class of degenerate second order differential operators in the space $C(S_d)$, where $S_d$ is the canonical simplex of $R^d$. The semigroups arise from the theory of Fleming--Viot processes in population genetics.

We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet–Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a Fréchet–Schwartz space does so in a special way. We single out this type of “rapid convergence” for a sequence of operators and study its relationship to the structure of the underlying space. Its relevance for Schauder decompositions and the connection to mean ergodic operators on Fréchet– Schwartz spaces is also investigated.

Let $X$ be a separable, infinite--dimensional real or complex Fréchet space admitting a continuous norm. Let ${v_n: ngeq 1}$ be a dense set of linearly independent vectors of $X$. We show that there exists a continuous linear operator $T$ on $X$ such that the orbit of $v_1$ under $T$ is exactly the set ${v_n: ngeq 1}$. Thus, we extend a result of Grivaux for Banach spaces to the setting of non--normable Fr'echet spaces with a continuous norm. We also provide some consequences of the main result.

Every Köthe echelon Fréchet space X that is Montel and not isomorphic to a countable product of copies of the scalar field admits a power bounded continuous linear operator T such that I − T does not have closed range, but the sequence of arithmetic means of the iterates of T converges to 0 uniformly on the bounded sets in X. On the other hand, if X is a Fréchet space which does not have a quotient isomorphic to a nuclear Köthe echelon space with a continuous norm, then the sequence of arithmetic means of the iterates of any continuous linear operator T (for which $(1/n)T^n$ converges to 0 on the bounded sets) converges uniformly on the bounded subsets of X, i.e., T is uniformly mean ergodic, if and only if the range of I−T is closed. This result extends a theorem due to Lin for such operators on Banach spaces. The connection of Browder’s equality for power bounded operators on Fréchet spaces to their uniform mean ergodicity is exposed. An analysis of the mean ergodic properties of the classical Cesàro operator on Banach sequence spaces is also given.

Let $(A, D(A))$ be a densely defined operator on a Banach space $X$. Characterizations of when $(A, D(A))$ generates a $C_0$-semigroup on $X$ are known. The famous result of Lumer and Phillips states that it is so if and only if $(A, D(A))$ is dissipative and $rg(lambda I − A) subseteq X$ is dense in $X$ for some $lambda>0$. There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran–Kaminska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non-normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented.

Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(mathbb{R})$ by a translation of the Ornstein-Uhlenbeck operator is also given.

The spectrum and point spectrum of the Cesàro averaging operator $C$ acting on the Fréchet space $C^infty(R^+)$ of all $C^infty$ functions on the interval $[0,infty)$ are determined. We employ an approach via $C_0$-semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of $C$ is also investigated.

We exhibit examples of Fréchet Montel spaces $E$ which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2, they are of independent interest and show for example that the canonical inclusion between James spaces $J_p subset J_q, 1 < p < q < infty,$ is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual $J'_p$ of the James space $J_p$, and permits us to show that the Fréchet space $J_{p^+} = cap_{q>p}J_q$ has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces.

We study $omega$-regularity of the solutions of certain operators that are globally $C^infty$-hypoelliptic in the N-dimensional torus. We also apply these results to prove the global $omega$-regularity for some classes of sublaplacians. In this way, we extend previous work in the setting of analytic and Gevrey classes. Different examples on local and global $omega$-hypoellipticity are also given.

Banach spaces which are Grothendieck spaces with the Dunford-Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11.77-93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-spaces are again GDP-spaces. Also, every complete injective space is a GDP-space. For $pin {0}cup [1,infty)$ it is shown that the classical co-echelon spaces $k_p(V)$ and $K_p(ov{V})$ are GDP-spaces if and only if they are Montel. On the other hand, $K_infty(ov{V})$ is always a GDP-space and $k_infty(V)$ is a GDP-space whenever its (Fréchet) predual, i.e., the Kothe echelon space $lambda_1(A)$, is distinguished.

Idempotent copulae have been characterised, in an implicit form, in cite{Sem02}; here we look at a few well known classes of copulas, namely, Fréchet copulas, ordinal sums, Archimedean copulas, copulas of the type $C(u,v)=uvpm f(u),g(v)$ and a special subset of copulas represented through measure-preserving transformations, and characterise those among these classes that are idempotent.

We present criteria for determining mean ergodicity of $C_0$--semigroups of linear operators in a sequentially complete, locally convex Hausdorff space $X$. A characterization of reflexivity of certain spaces $X$ with a basis via mean ergodicity of equicontinuous $C_0$-semigroups acting in $X$ is also presented. Special results become available in Grothendieck spaces with the Dunford-Pettis property.

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kuhnemund. We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in $C_b(R^N)$ and to evolution operators associated with nonautonomous second-order differential operators in $C_b(R^N)$ with time-periodic coefficients.

A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator C acting on the weighted Banach sequence space $c_0(w)$ for a bounded, strictly positive weight $w$. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which are not present in the classical setting of $c_0$.

For $C_0$-semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the infinitesimal generator A being a closed subspace or, whether 0 belongs to the resolvent set of A or is a simple pole of the resolvent map $lambdamapsto (lambda − A)^{-1}$. It is shown in the setting of locally convex spaces (even in Fréchet spaces), that neither of these criteria remain equivalent to uniform ergodicity of the semigroup (i.e., the averages should now converge for the topology of uniform convergence on bounded sets). Our aim is to exhibit new results dealing with uniform mean ergodicity of $C_0$-semigroups in more general spaces. A characterization of when a complete, barrelled space with a basis is Montel, in terms of uniform mean ergodicity of certain C0-semigroups acting in the space, is also presented.

Aspects of the theory of mean ergodic operators and bases in Fréchet spaces were recently developed in [A.A. Albanese, J. Bonet, W.J. Ricker, Mean ergodic operators i Fréchet spaces, Ann. Acad. Sci. Math. Fenn. Math. 34 (2009), 1-36]. This investigation is extended here to the class of barrelled locally convex spaces. Duality theory, also for operators, plays a prominent role.

Various properties of the (continuous) Cesàro operator C, acting on Banach and Fréchet spaces of continuous functions and $L_p$-spaces, are investigated. For instance, the spectrum and point spectrum of C are completely determined and a study of certain dynamics of C is undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of C acting in the various spaces is identified.

In this paper we prove the global $C^infty$ and Gevrey hypoellipticity on the multidimensional torus for some classes of degenerate elliptic operators.

The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators $A^{gamma,b}u(x) = gamma xu''(x) + bu'(x)$, with constants $ gamma > 0$ and $ bgeq 0$, in the space $C([0,infty])$.

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, $WF_*$, of classical distributions. In particular, we have the following inclusion $$ WF_ast (u)subset WF_ast(Pu)cup Sigma, quad uinD^prime(Omega), $$ where $Omega$ is an open subset of $R^n$, $P$ is a linear partial differential operator with coefficients in a suitable ultradifferentiable class, and $Sigma$ is the characteristic set of $P$. Some applications are also investigated.

We prove some permanence results with respect to quotient spaces and to projective and injective tensor products of the Dunford--Pettis and Grothendieck properties in the setting of locally convex Hausdorff spaces.

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $C$ when acting on the weighted Banach sequence spaces $ell_p(w)$, $1<p<infty$, for a positive, decreasing weight $w$, thereby extending known results for $C$ when acting on the classical spaces $ell_p$. New features arise in the weighted setting (e.g., existence of eigenvalues, compactness) which are not present in $ell_p$.

The Ces`aro operator $C$, when acting in the classical growth Banach spaces $A^{-gamma}$ and $A^{-gamma}_0$ , for $gamma>0$, of analytic functions on $D$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of $C$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $D$ which $C$ maps into $A^{-gamma}$ (resp. into $A^{-gamma}_0$); this optimal domain space always contains $A^{-gamma}$ (resp. $A^{-gamma}_0$ ) as a proper subspace.

The classical spaces $ell^{p+}$, $1leq p<infty$, and $L^{p−}$, $1<pleqinfty$, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ${mathbb C}^{mathbb N}$, $L^p_{loc}({mathbb R}^+)$ for $1<p<infty$ and $C({mathbb R}^+)$, which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

The spectrum of the Cesàro operator C, which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fréchet spaces or (LB)-spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, nevr nuclear. Some consequences concerning the mean ergodicity of C are deduced.

The discrete Cesàro operator $C$ is investigated in the class of power series spaces $Lambda_0(alpha)$ of finite type. Of main interest is its spectrum, which is distinctly different in the cases when $Lambda_0(alpha)$ is nuclear and when it is not. Actually, the nuclearity of $Lambda_0(alpha)$ is characterized via certain properties of the spectrum of $C$. Moreover, $C$ is always power bounded, uniformly mean ergodic, and whenever $Lambda_0(alpha)$ is nuclear, also $(lambda- C)^m(Lambda_0(alpha))$ is closed in $Lambda_0(alpha)$ for each $minmathbb{N}$.

The Banach spaces $ces(p)$, $1<p<infty$, were intensively studied by G. Bennett and others. The largest solid Banach lattice in $mathbb{C}^{mathbb {N}}$ which contains $ell_p$ and which the Cesàro operator $C colonmathbb{C}^{mathbb {N}}tomathbb{C}^{mathbb {N}}$ maps into $ell_p$ is $ces(p)$. For each $1leq p<infty$, the (positive) operator $C$ also maps the Fréchet space $ell_{p+}=cap_{q>p}ell_q} into itself. It is shown that the largest solid Fréchet lattice in $mathbb{C}^{mathbb {N}$ which contains $ell_{p+} and which $C$ maps into $ell_{p+} is precisely $ces(p+) :=cap_{q>p}ces(q)$. Although the spaces $ell_{p+}$ are well understood, it seems that the spaces $ces(p+)$ have not been considered at all. A detailed study of the Fréchet spaces $ces(p+)$, $1leq p<infty$, is undertaken. They are very different to the Fréchet spaces $ell_{p+}$ which generate them in the above sense. We prove that each $ces(p+)$ is a power series space of finite type and order one, and that all the spaces $ces(p+)$, $1leq p<infty$, are isomorphic.

Let $(T(t))_{tgeq 0}$ be a strongly continuous $C_0$-semigroup of bounded linear operators on a Banach space $X$ such that $lim_{ttoinfty}||T(t)||/t = 0$. Characterizations of when $(T(t))_{tgeq 0}$ is uniformly mean ergodic, i.e., of when its Cesàro means $r^{-1}int_0^r T(s)ds$ converge in operator norm as $to infty$, are known. For instance, this is so if and only if the infinitesimal generator $A$ has closed range in $X$ if and only if $lim_{lambdato 0^+}lambda R(lambda, A)$ exists in the operator norm topology (where $R(lambda,A)$ is the resolvent operator of $A$ at $lambda$). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of $C_0$-semigroups in particular Fréchet function and sequence spaces are presented.

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operator T to the operator norm convergence of certain sequences of operators generated by T, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

We prove the following inclusion [ WF_* (u)subset WF_*(Pu)cup Sigma, quad uinE^prime_ast(Omega), ] where $WF_*$ denotes the non--quasianalytic Beurling or Roumieu wave front set, $Omega$ is an open subset of $R^n$, $P$ is a linear partial differential operator with suitable ultradifferentiable coefficients, and $Sigma$ is the characteristic set of $P$. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.